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Axel D. Becke develops practical approximations for density functional theory. His 1988 exchange functional uses the local electron density and its gradient to improve on a purely local density approximation. Its construction imposes the proper long-distance behavior of the exchange-energy density and fits one parameter using atomic exchange energies. This supplies a concrete example of building a useful approximate energy functional from physical constraints and reference calculations. The asymptotic statement concerns an energy density, and does not automatically establish the asymptotic behavior of its functional derivative.^[ref]

Computational chemistry uses numerical models to predict or interpret chemical structure, energetics and dynamics. Electronic calculations may approximate the Schrödinger equation through Hartree-Fock theory or density functional theory; molecular simulations can instead evolve nuclei using an approximate potential.

A geometry optimization searches for a stationary point on a potential energy surface, while a vibration calculation examines its local curvature. These are different tasks: a converged numerical solution need not be an accurate physical prediction, and a stationary structure need not be the global energy minimum. Method, basis, environment and uncertainty therefore belong with a reported computed result.

Electronic structure describes the quantum states and spatial distribution of a system's electrons, including their energies, occupancies and correlations. It determines much of the chemical bonds, optical response and reactivity of atoms, molecules and solids.

An electron configuration is one compact approximate description, but it is not the full many-electron state. In molecular calculations, molecular orbitals and electron density reveal where charge is distributed; a many-electron wave function also encodes correlations. In a crystal, energy bands express how allowed electronic energies depend on wavevector. Approximations such as Hartree-Fock theory and density functional theory address different representations of the same underlying quantum problem.

A generalized gradient approximation models the exchange and correlation contribution in density functional theory using both the local electron density $n(\mathbf{r})$ and its spatial gradient $\nabla n(\mathbf{r})$. Its schematic form is $E_{\mathrm{xc}}[n]=\int f(n,\nabla n)d^{3}r$. It extends the local density approximation, which uses only the density at each point.

Two regions with equal density can therefore receive different energy contributions if one is nearly uniform and the other varies rapidly. This makes the approximation useful for chemical bonds, surfaces and molecular tails. The improvement is still approximate: adding density gradients does not automatically recover every long-distance interaction or the exact exchange potential. The Becke 1988 exchange functional is an example of a gradient-dependent exchange model.

John E. McGrady studies how electronic structure controls the chemical bonds and magnetism of inorganic metal clusters. In an endohedral metal cluster, a metal atom sits inside a cage; calculating the interaction between that atom and the cage can distinguish a nearly independent enclosed metal from a structure with strongly shared electron density. His group uses density functional theory and multiconfigurational methods to examine these possibilities.^[ref]

Homepage: mcgrady.chem.ox.ac.uk/

Lu Jeu Sham and Walter Kohn developed self-consistent equations that make density functional theory computationally useful. The Kohn-Sham equations introduce an auxiliary system of noninteracting electrons with the same ground-state electron density as the interacting system. An effective potential includes the external potential, electrostatic repulsion and an exchange-correlation functional. Solving the equations and updating the electron density iteratively gives a self-consistent state. The difficult many-electron effects remain in the exchange-correlation functional, so practical calculations still need an approximation to it.^[ref]

Density functional theory replaces the interacting many-electron ground state wavefunction by its spatial density $n(\mathbf{r})$. For fixed interaction and a nondegenerate ground state, the Hohenberg-Kohn theorem fixes the external potential up to an additive constant. If two distinct potentials gave the same density, their strict variational energy inequalities would sum to the contradiction $E_0+E_0^{\prime }<E_0+E_0^{\prime }$.

The density functional variational principle minimizesover admissible densities of the correct particle number. The universal $F$ contains the kinetic and interaction energies. This is exact in principle; an explicit useful functional is the practical difficulty.^[ref]

The Kohn-Sham equations introduce noninteracting orbitals with the same density and decompose $F=T_s+E_H+E_{\mathrm{xc}}$. With $n={\sum }_i{f}_i|{\varphi }_i{|}^2$,The exchange-correlation functional includes exchange, correlation and the interacting kinetic-energy correction relative to $T_s$. Guess a density, construct the potentials, solve and occupy orbitals, then mix and iterate to self-consistent field convergence. The local density approximation uses $E_{\mathrm{xc}}^{\mathrm{LDA}}=\int n{ϵ}_{\mathrm{xc}}^{\mathrm{uniform}}(n)d^{3}r$; a generalized gradient approximation also depends on density gradients.

Kohn–Sham orbital energies are not generally excitation energies, and their sum double counts interactions. The Kohn-Sham energy double-counting correction iswith ion–ion energy added for a total crystal energy. Approximate functionals can miss self-interaction cancellation, band gaps and strong correlation; convergence alone does not prove the global ground state.^[ref]

Kohn-Sham equations in density functional theory use an auxiliary one-electron Hamiltonian whose effective potential depends on the density. Its orbitals construct a density representing an interacting electronic system; this is distinct from explicitly diagonalizing a large many-electron determinant Hamiltonian.^[ref]

The lower curve is a valence band and the upper curve is a conduction band, separated by an indirect band gap. The readable upper curve has a minimum just inside the X boundary and slightly below the plotted chemical potential. It is therefore partially occupied, while the visible valence states lie below the chemical potential and are filled at low temperature. The sample is electron-doped, with electron pockets around the six symmetry-related conduction minima of silicon.

The full high-symmetry path and part of the lower-left dispersion are absent from the held image. Source gap: those missing path labels and curve segments cannot be read or reconstructed from this copy. The identification of the conventional six silicon valleys and the valence maximum at $\Gamma$ uses the usual silicon band model, rather than purported readings of missing pixels.

Two methods for calculating such an electronic band structure are density functional theory, solving a self-consistent periodic electronic problem, and a tight-binding model, diagonalizing a hopping Hamiltonian built from localized atomic orbitals. Neither is a measurement of the plotted dispersion relation.

A theoretical condensed matter physicist who, with Walter Kohn, established a foundation of density functional theory in 1964. Their variational principle expresses the ground-state energy of interacting electrons through the electron density, making a three-dimensional quantity central to a many-particle problem.^[ref]

His 1967 work also ruled out the relevant finite-temperature long-range order in one- and two-dimensional bosonic and fermionic systems under the theorem's assumptions. This helps explain why dimensionality matters when discussing Bose-Einstein condensation.^[ref]

S. Bendjaballah coauthored research on density functional theory.^[ref]

A framework describing electronic ground-state properties using the electron density. Practical approximations require an exchange-correlation functional; this does not mean that explicit electrons have been replaced by a classical atom-only force field.

Walter Kohn and Pierre Hohenberg established a foundation for density functional theory: under their ground-state assumptions, the electron density determines the external scalar potential up to an additive constant. They formulated the ground-state energy as a universal density functional plus the external-potential contribution, with a variational principle selecting the physical density. This replaces the full many-electron wave function as the basic variable by a function of only three spatial coordinates. The theorem establishes the framework, rather than providing an explicit exact functional for practical computation.^[ref]